In the first lesson of chapter 4, the measure of angles in a circle was introduced. We find out how to measure angles by revolutions, radians, and degrees. When measuring an angle go in a counter clock wise rotation. If you would want to change units from radian to Degree, then multiply the number that you want to change by 180°/pi radians. If you wanted to change from degree to radian then multiply by pi radians/180°. If you want to change from revolutions to radians then divide by 2pi.
robert w
In lesson 4-1 we learned that by using revolutions, radians, and degrees we can measure angles and arcs. We also learned how to convert radians to degrees. Pretend that we have 31.4159 radians and we want to convert that to degrees. We need to take 31.4159 and multiple it by 180/pi or 360/2pi and we would get 1800 degrees. If you want to measure an angle you go in a clockwise rotation. If you want to go from revolutions to radians then you must divide by 2pi.
jesse b
4-2: Lengths of Arcs and Areas of Sectors A couple of tips for this lesson would be:
1.) Do not approximate the answer except at the really end!
2.) Whatever you want goes on top!
3.) Radians are the units of measure of the central angle in the Circular Sector Area Formula!
-Kelly H.
In Lesson 2 we learned how to find the arc length of any given central angle using the formula s=rθ. theta representing the central angle in radians and r being the radius of the circle. We also learned how to find the area of the part of the a disk that is on or in the interior of a central angle using the Circular Sector Area Formula. This formula is A= 1/2θr^2. A representing the area, r representing the radius, and theta representing the measure of the central angle in radians.
-Allison R
4-3: Sines, Cosines, and Tangents
Vocab:The Unit Circle - A circle whose center is on the origin and radius is 1 unit. Ordered Pair on the Unit Circle - For all real numjbers Ø, (cos Ø, sin Ø) is the image of the point (1,0) under a rotation of magnitude Ø about the origin. That is, (cos Ø, sin Ø) = RØ(1,0).
-Austin A.
Lesson 3 is about Sines, Cosines and Tangents. This lesson talks about the unit circle. The unit circle is the circle with its center at (0,0) and a radius of 1. The cosine (cos) of theta is the x-coordinate, and the sine (sin) of theta is the y-coordinate. At (1,0) on the unit circle cos is 0 radians and 0° and the sin is 0 radians and 0°. At (0,1) on the unit circle cos is pi/2 and 90° and sin is pi/2 and 90°. At (-1,0) on the unit circle cos is pi and 180° and sin is pi and 180°. At (0,-1) on the unit circle cos is 3pi/2 and 270° and sin is 3pi/2 and 270°. The definiton of tangent is for all real numbers theta, cos cannot equal 0 then tan theta equals sin theta/cos theta. When cos 0 the tan is undefined.
4-4: Basic Identities Involving Sines, Cosines, and Tangents
Vocab:
Identity-an equation that is true for all values of the variables for which the expressions are defined
Pythagorean identity-for every theta, cos^2 theta + sin^2 theta= 1
Opposite theorem- for all theta, cos(-theta) equals cos theta, sin (-theta) equals -sin theta, and tan (-theta) equals -tan theta
Supplements Theorem- for all theta in radians, sin (pie - theta) equalssin theta, cos (pie - theta) equals -cos theta, tan (pie - theta) equals -tan theta
Complements Theorem- for all theta in radians, sin (pie/2 - theta) equals cos theta and cos (pie/2 - theta) equals sin theta
Half-turn Theorem- for all theta in radians, cos (cos + theta) equals -cos theta, sin (pie + theta) equals -cos theta, tan (pie + theta) equals tan theta
This chapter is about how you can find certain angles, shapes, and equations within the circles.
Zachary Paul L.
Yea boy!
The first theorem of lesson 4-4 is the Pythagorean Identity, stating that the cos. of a theta squared + sin. of the theta squared =1. From this, you can find the cos. or sin. of a theta if you know the other. The Opposites Theorem shows what cos., sin., and tan. are in a theta's opposite compared to the original theta. The Supplements Theorem shows the relation between the sin., cos., and tan. of a theta and the sin., cos., and tan., of Pi - theta. The Complements Theorem shows the relation between sin., cos., and tan., of a theta and sin., cos., and tan. of Pi/2 - theta. The Half Turn Theorem shows the relation in sin., cos., and tan values between a theta and when adding Pi to the theta.
-Nathan F.
This chapter is about the functions of sine, cosine, and tangents and how to graph them.
The sine function is a function that maps each real theta to the y-coordinate of the image of (1,0) under a rotation of theta. Basically, this means that for every theta, such as pi over 2, you take the sine coordinate or y. For pi over 2, sine would be 1 because the coordinate for pi over 2 is (0,1). To graph sine, start at (0,0) because on the unit circle, the first theta is 0, and the first sine coordinate is zero. Just follow the thetas and their corresponding sine coordinates and graph. This should give you the graph of sine. The cosine function is almost exactly the same as sine exept it starts at (0,1). This is the starting point on the unit circle. The same thing applies to cosine as sine when graphing. Use the theta and corresponding cosine coordinates. Last but not least tangent function. Tangent starts at (0,0) but it graphed differently. Tangent = sine/cosine, so take the sine value, starting at (0,0) on the unit circle and insert the respective coordinate. The graph looks totally different, and it should. Some theorems that may help in this process of graphing are the definition of a period-which is simply the length it takes to restart the pattern. For sine and cosine, that pattern is the graph going up to 1 and down to -1, then ending at zero. The Periodicity Theorem states that for all theta, and for every integer:
sin(theta + 2Pin) = sin of theta
cos(theta + 2Pin) = cos of theta
tan(theta + Pin) = tan of theta
This means that since sine and cosine restart their patter at every 2Pi, adding 2Pi to any number on the sine function will take you to the exact number on the function either 2Pi or -2Pi. The same for tangent except tangent's period is only Pi.
-ANDREW C.
4-7: Scale-Change Images of Circular Functions
The main focus of lesson 7 is the scale-change images of circular functions!
The pitch of the tone is related to the period of the wave; the longer the period, the lower the pitch.
The intensity of the tone is related to the ampitude of the wave.
Stretching a sine wave horizontally or vertically pictures changes in a tone's pitch and intensity.
These stretches can be accomplished by applying the Graph Scale-Change Theorem.
And just a reminder of what the graph scale-change theorem is:
In a relation described by a sentence in x and y, the following two processes yield the same graph:
(1) replacing x by x/a and y by y/b in the sentence;
(2) applying the scale change (x,y)->(ax,by) to the graph of the original relation.
This lesson can be summed up in one simple word: Translation. We have all done it before. We have translated graphs in in previous chapters. A translation that shifts the graph on the horizontal axis is called a phase shift. This is the greatest negative or the least positive value. A phase shift may cause a sine graph to be mapped onto a cosine graph or visa versa.
Katie S.
Translation.
But besides that, dealing with sin and cos graphs a horizontal translation is called a phase shift. A phase shift is the least positive or greatest negative value in a horizontal translation for a sin or cos wave. Because of the phase shift it will be mapped onto itself.
Robert W.
4-9: The Graph-Standardization Theorem
In this lesson, we learned how to apply a translation and scale change to a graph of the sine or cosine function at the SAME TIME!
One of the main things I got out of this lesson was how to moves the axes in order to accommodate the scale change and translation.
Example: y=3sin(¼x - π) +3
In order to do this, the first step is to find the amplitude. This can always be found in front of the equation. So, amp=3
Next, we need to find the vertical shift which is usually found at the end outside of the parentheses. vert shift=3
The maximum and minimum can be found by taking the positive and negative amplitude and adding the vertical shift to it. max=3+3π=6
min=-3+3=0
The period is 2π times the scale change (the reciprocal of what is in the parentheses.) period=(2π)4= 8π
The phase shift is the opposite of what is in the parentheses times the scale change. phase shift= (π)4= 4π
The domain can be found by adding the phase shift to 0, which in this case is 4π. 4π plus one period (8π) is 12π. So the domain is
4π≤ x ≤ 12π.
The range is obviously the minimum and maximum values. So the range is 0 ≤ y ≤ 6
With all of this information, the graph is really easy to put on paper.
The x-axis is moved to the value of 0 on the y-axis.
Graph-Standardization Theorem: x replaced by x-h/a and y replaced by y-k/b.
S(x,y)= (ax, by)
T(x,y)= (x+h, y+k)
Andrea M.
4-10: Modeling with Circular Functions
Section 4-10 explained how to model phenomena (most of which are periodic) whose graphs are sine waves. To put this into simpler terms, this lesson taught us how to find a sine wave of "best-fit" for a series of data points.
The easiest way to find the sine wave of "best-fit" is to first enter the data into your graphing calculator. To do so from the main screen, click STAT and and press ENTER (1: EDIT). You can list the x-values and y-values in any "L" column as long as you remember which one is which. For the sake of this lesson let's say enter the x-values in L1 and the y-values in L2. Then press 2ND MODE to return to the main screen. Next, press STAT, move right to CALC, and scroll down to C: SinReg; press Enter. Follow by keying, 2ND 1 (L1); ,(COMMA); 2ND 2; , (COMMA); VARS, Y-VARS, and ENTER. Your screen should now look like this: SinReg L1,L2,Y1. Now, press ENTER. An equation should show that is in the form of a sine wave. To see the whole equation, click on the Y= button.
To graph the model you have just found, turn on your stat plot. Hit 2ND Y=, ENTER, and click ON. After this press the GRAPH button. You will see a sine wave connect your different data points. From this point you can do a variety of calculations by TRACING the graph and by using ZOOM.
Now you are all set to find a model to fit a sine wave and graph it.
Table of Contents
Circular Functions
4-1: Measures of Angles and Rotations
In the first lesson of chapter 4, the measure of angles in a circle was introduced. We find out how to measure angles by revolutions, radians, and degrees. When measuring an angle go in a counter clock wise rotation. If you would want to change units from radian to Degree, then multiply the number that you want to change by 180°/pi radians. If you wanted to change from degree to radian then multiply by pi radians/180°. If you want to change from revolutions to radians then divide by 2pi.robert w
In lesson 4-1 we learned that by using revolutions, radians, and degrees we can measure angles and arcs. We also learned how to convert radians to degrees. Pretend that we have 31.4159 radians and we want to convert that to degrees. We need to take 31.4159 and multiple it by 180/pi or 360/2pi and we would get 1800 degrees. If you want to measure an angle you go in a clockwise rotation. If you want to go from revolutions to radians then you must divide by 2pi.
jesse b
4-2: Lengths of Arcs and Areas of Sectors A couple of tips for this lesson would be:
1.) Do not approximate the answer except at the really end!2.) Whatever you want goes on top!
3.) Radians are the units of measure of the central angle in the Circular Sector Area Formula!
-Kelly H.
In Lesson 2 we learned how to find the arc length of any given central angle using the formula s=rθ. theta representing the central angle in radians and r being the radius of the circle. We also learned how to find the area of the part of the a disk that is on or in the interior of a central angle using the Circular Sector Area Formula. This formula is A= 1/2θr^2. A representing the area, r representing the radius, and theta representing the measure of the central angle in radians.
-Allison R
4-3: Sines, Cosines, and Tangents
Vocab: The Unit Circle - A circle whose center is on the origin and radius is 1 unit.Ordered Pair on the Unit Circle - For all real numjbers Ø, (cos Ø, sin Ø) is the image of the point (1,0) under a rotation of magnitude Ø about the origin. That is, (cos Ø, sin Ø) = RØ(1,0).
-Austin A.
Lesson 3 is about Sines, Cosines and Tangents. This lesson talks about the unit circle. The unit circle is the circle with its center at (0,0) and a radius of 1. The cosine (cos) of theta is the x-coordinate, and the sine (sin) of theta is the y-coordinate. At (1,0) on the unit circle cos is 0 radians and 0° and the sin is 0 radians and 0°. At (0,1) on the unit circle cos is pi/2 and 90° and sin is pi/2 and 90°. At (-1,0) on the unit circle cos is pi and 180° and sin is pi and 180°. At (0,-1) on the unit circle cos is 3pi/2 and 270° and sin is 3pi/2 and 270°. The definiton of tangent is for all real numbers theta, cos cannot equal 0 then tan theta equals sin theta/cos theta. When cos 0 the tan is undefined.
Travis D.
4-4: Basic Identities Involving Sines, Cosines, and Tangents
Vocab:Identity-an equation that is true for all values of the variables for which the expressions are defined
Pythagorean identity-for every theta, cos^2 theta + sin^2 theta= 1
Opposite theorem- for all theta, cos(-theta) equals cos theta, sin (-theta) equals -sin theta, and tan (-theta) equals -tan theta
Supplements Theorem- for all theta in radians, sin (pie - theta) equalssin theta, cos (pie - theta) equals -cos theta, tan (pie - theta) equals -tan theta
Complements Theorem- for all theta in radians, sin (pie/2 - theta) equals cos theta and cos (pie/2 - theta) equals sin theta
Half-turn Theorem- for all theta in radians, cos (cos + theta) equals -cos theta, sin (pie + theta) equals -cos theta, tan (pie + theta) equals tan theta
This chapter is about how you can find certain angles, shapes, and equations within the circles.
Zachary Paul L.
Yea boy!
The first theorem of lesson 4-4 is the Pythagorean Identity, stating that the cos. of a theta squared + sin. of the theta squared =1. From this, you can find the cos. or sin. of a theta if you know the other. The Opposites Theorem shows what cos., sin., and tan. are in a theta's opposite compared to the original theta. The Supplements Theorem shows the relation between the sin., cos., and tan. of a theta and the sin., cos., and tan., of Pi - theta. The Complements Theorem shows the relation between sin., cos., and tan., of a theta and sin., cos., and tan. of Pi/2 - theta. The Half Turn Theorem shows the relation in sin., cos., and tan values between a theta and when adding Pi to the theta.
-Nathan F.
4-5: Exact Values of Sines, Cosines, and Tangents
Katie K?4-6: The Sine, Cosine, and Tangent Functions
This chapter is about the functions of sine, cosine, and tangents and how to graph them.
The sine function is a function that maps each real theta to the y-coordinate of the image of (1,0) under a rotation of theta. Basically, this means that for every theta, such as pi over 2, you take the sine coordinate or y. For pi over 2, sine would be 1 because the coordinate for pi over 2 is (0,1). To graph sine, start at (0,0) because on the unit circle, the first theta is 0, and the first sine coordinate is zero. Just follow the thetas and their corresponding sine coordinates and graph. This should give you the graph of sine. The cosine function is almost exactly the same as sine exept it starts at (0,1). This is the starting point on the unit circle. The same thing applies to cosine as sine when graphing. Use the theta and corresponding cosine coordinates. Last but not least tangent function. Tangent starts at (0,0) but it graphed differently. Tangent = sine/cosine, so take the sine value, starting at (0,0) on the unit circle and insert the respective coordinate. The graph looks totally different, and it should. Some theorems that may help in this process of graphing are the definition of a period-which is simply the length it takes to restart the pattern. For sine and cosine, that pattern is the graph going up to 1 and down to -1, then ending at zero. The Periodicity Theorem states that for all theta, and for every integer:
sin(theta + 2Pin) = sin of theta
cos(theta + 2Pin) = cos of theta
tan(theta + Pin) = tan of theta
This means that since sine and cosine restart their patter at every 2Pi, adding 2Pi to any number on the sine function will take you to the exact number on the function either 2Pi or -2Pi. The same for tangent except tangent's period is only Pi.
-ANDREW C.
4-7: Scale-Change Images of Circular Functions
The main focus of lesson 7 is the scale-change images of circular functions!
The pitch of the tone is related to the period of the wave; the longer the period, the lower the pitch.
The intensity of the tone is related to the ampitude of the wave.
Stretching a sine wave horizontally or vertically pictures changes in a tone's pitch and intensity.
These stretches can be accomplished by applying the Graph Scale-Change Theorem.
And just a reminder of what the graph scale-change theorem is:
In a relation described by a sentence in x and y, the following two processes yield the same graph:
(1) replacing x by x/a and y by y/b in the sentence;
(2) applying the scale change (x,y)->(ax,by) to the graph of the original relation.
-Kelly H.
4-8: Translation Images of Circular Functions
This lesson can be summed up in one simple word: Translation. We have all done it before. We have translated graphs in in previous chapters. A translation that shifts the graph on the horizontal axis is called a phase shift. This is the greatest negative or the least positive value. A phase shift may cause a sine graph to be mapped onto a cosine graph or visa versa.Katie S.
Translation.
But besides that, dealing with sin and cos graphs a horizontal translation is called a phase shift. A phase shift is the least positive or greatest negative value in a horizontal translation for a sin or cos wave. Because of the phase shift it will be mapped onto itself.
Robert W.
4-9: The Graph-Standardization Theorem
In this lesson, we learned how to apply a translation and scale change to a graph of the sine or cosine function at the SAME TIME!One of the main things I got out of this lesson was how to moves the axes in order to accommodate the scale change and translation.
Example: y=3sin(¼x - π) +3
In order to do this, the first step is to find the amplitude. This can always be found in front of the equation. So, amp=3
Next, we need to find the vertical shift which is usually found at the end outside of the parentheses. vert shift=3
The maximum and minimum can be found by taking the positive and negative amplitude and adding the vertical shift to it. max=3+3π=6
min=-3+3=0
The period is 2π times the scale change (the reciprocal of what is in the parentheses.) period=(2π)4= 8π
The phase shift is the opposite of what is in the parentheses times the scale change. phase shift= (π)4= 4π
The domain can be found by adding the phase shift to 0, which in this case is 4π. 4π plus one period (8π) is 12π. So the domain is
4π≤ x ≤ 12π.
The range is obviously the minimum and maximum values. So the range is 0 ≤ y ≤ 6
With all of this information, the graph is really easy to put on paper.
The x-axis is moved to the value of 0 on the y-axis.
Graph-Standardization Theorem: x replaced by x-h/a and y replaced by y-k/b.
S(x,y)= (ax, by)
T(x,y)= (x+h, y+k)
Andrea M.
4-10: Modeling with Circular Functions
Section 4-10 explained how to model phenomena (most of which are periodic) whose graphs are sine waves. To put this into simpler terms, this lesson taught us how to find a sine wave of "best-fit" for a series of data points.The easiest way to find the sine wave of "best-fit" is to first enter the data into your graphing calculator. To do so from the main screen, click STAT and and press ENTER (1: EDIT). You can list the x-values and y-values in any "L" column as long as you remember which one is which. For the sake of this lesson let's say enter the x-values in L1 and the y-values in L2. Then press 2ND MODE to return to the main screen. Next, press STAT, move right to CALC, and scroll down to C: SinReg; press Enter. Follow by keying, 2ND 1 (L1); ,(COMMA); 2ND 2; , (COMMA); VARS, Y-VARS, and ENTER. Your screen should now look like this: SinReg L1,L2,Y1. Now, press ENTER. An equation should show that is in the form of a sine wave. To see the whole equation, click on the Y= button.
To graph the model you have just found, turn on your stat plot. Hit 2ND Y=, ENTER, and click ON. After this press the GRAPH button. You will see a sine wave connect your different data points. From this point you can do a variety of calculations by TRACING the graph and by using ZOOM.
Now you are all set to find a model to fit a sine wave and graph it.
Zach G.